Uniformity and Loewner Condtions of Metric Spaces

Abstract

In this paper, we establish four equivalent conditions in the metric space setting. These conditions concern the (inner) uniformity of metric spaces, and (locally) Loewner conditions, weak slice conditions and k-cap conditions of metric measure spaces. This investigation completes the related study started by Bonk, Heinonen, and Koskela in 2001. Also, two examples are constructed to demonstrate that the assumption of the regularity of the underlying spaces can not be removed.

1 Introduction

A metric space (X, d) is called uniform if every pair of points in X can be joined by a so called uniform curve (see Sect. 2.1 below for the precise definition). If a domain, i.e., an open and connected subset, in X satisfies this condition, then it is called a uniform domain. The concept of uniformity in the metric space setting was introduced by Bonk et al. [2]. Uniform domains in ${{\mathbb {R}}}^n$ with $n\ge 2$ were introduced by Martio and Sarvas [19]. Since its introduction, this concept has played a significant role in the study of geometric function theory, see [16] and references therein.

In order to study the theory of quasiconformal mappings in metric spaces, Heinonen and Koskela introduced a class of metric spaces which is called Loewner spaces [12]. A metric measure space is called Loewner if it satisfies a Loewner condition (see Definition 2.2 below for the details). All Euclidean spaces ${{\mathbb {R}}}^n$ with $n\ge 2$ are Loewner. See [4, 12, 18] for more examples of Loewner spaces.

In 2001, Bonk, Heinonen, and Koskela considered the relation between uniformity and Loewner condition, and established the following sufficient condition for a metric measure space to be Loewner.

Theorem A

[2, Theorem 6.4]. Suppose that $(X,d,\mu )$ is a locally compact and noncomplete metric measure space. If $(X,d,\mu )$ is a uniform and locally Q-Loewner space with $Q>1$, then it is Q-Loewner. This implication is quantitative.

See Chapters 6 and 7 in [2] for the significance and the importance of Theorem A. Throughout the paper, for a metric measure space $(X,d,\mu )$, we assume that the measure $\mu$ is locally finite and Borel regular with dense support. The terminology in Theorem A and in the rest of this section will be introduced in the second section unless otherwise stated.

Here and hereafter, if a condition $\varpi$ with data $\chi$ implies a condition $\varpi '$ with data $\chi '$ so that $\chi '$ depends only on $\chi$ and other given quantities, then we say that $\varpi$ implies $\varpi '$ quantitatively, or the implication is quantitative. If also $\varpi '$ implies $\varpi$ quantitatively, then we say that $\varpi$ and $\varpi '$ are quantitatively equivalent, or the equivalence is quantitative.

In [22], Yang constructed an $L_{p}^{1}$ domain D in ${{\mathbb {R}}}^n$ for all $p\ge 1$, which is not uniform; see [22, Theorem 1.1]. Moreover, it is known that every $L_{n}^{1}$ domain is $\textrm{QED}$ (the abbreviated form of the phrase “quasiextremal distance”) (cf. [17]), and every $\textrm{QED}$ domain is Loewner (cf. [3]). These facts guarantee that the domain D constructed in [22] is Loewner. Since D is not uniform, we conclude that the converse of Theorem A is invalid. Naturally, one will ask if there is any condition which, by combining with the Loewner condition, constitutes a necessary and sufficient condition for a metric measure space to be uniform and locally Loewner. The purpose of this paper is to study this question, and the following are our answers.

Theorem 1.1

Suppose that $(X,d,\mu )$ is Q-regular with $Q>1$, and $G\subseteq X$ is a locally compact and c-quasiconvex domain with $c\ge 1$, where the case $G=X$ is included when (X, d) is noncomplete. Then the following are quantitatively equivalent:

$({\mathfrak {a}})$:

G is uniform and locally Q-Loewner;

$({\mathfrak {b}})$:

G is Q-Loewner and inner uniform;

$({\mathfrak {c}})$:

G is Q-Loewner and satisfies a weak slice condition;

$({\mathfrak {d}})$:

G is Q-Loewner and satisfies a k-cap condition.

We shall construct two examples, i.e., Examples 5.1 and 5.2 below, to demonstrate that the assumption on the regularity of the underlying spaces in Theorem 1.1 cannot be weakened to the one of being doubling, and thus, this implies that such an assumption cannot be removed from Theorem 1.1.

The rest of this paper is arranged as follows. In Sect. 2, necessary preliminaries will be introduced and two lemmas will be proved. Several auxiliary lemmas will be shown in Sect. 3. Section 4 will be devoted to the proof of Theorem 1.1, and in Sect. 5, the aforementioned examples will be exhibited.

2 Preliminaries

Let (X, d) denote a metric space, ${\overline{X}}$ the metric completion of X, and $\partial X={\overline{X}}\backslash X$, the metric boundary of X. For a metric space (X, d), we write $|x-y|=d(x,y)$ for the distance between two pints x and y in X. The symbol ${{\mathbb {B}}}(x,r)$ denotes the open ball in X with center x and radius $r>0$, i.e., ${{\mathbb {B}}}(x,r)=\{y\in X:\; |y-x|<r\}$. For $\lambda >0$, define $\lambda {{\mathbb {B}}}(x,r)={{\mathbb {B}}}(x,\lambda r)$. When working in a given domain $G\varsubsetneq X$, for any $x\in G$, let $\lambda {{\mathbb {B}}}(x)=\lambda {{\mathbb {B}}}(x,d_G(x))$, where $d_G(x)$ denotes the distance of x to the boundary $\partial G$ of G. Observe that every ball $\lambda {{\mathbb {B}}}(x)$ in ${{\mathbb {R}}}^n$ is contained in G whenever $x\in G$ and $\lambda \in (0, 1)$. But in the metric space setting, this property is no longer valid. To overcome this deficiency, we introduce a new notation $\lambda {{\mathbb {B}}}_G(x)=\lambda {{\mathbb {B}}}(x)\cap G$, which is called the Whitney ball at x with parameter $\lambda \in (0,1)$. Notice that not every Whitney ball is connected. Also, we use $\lambda {\overline{{{\mathbb {B}}}}}_G(x)$ to denote the closure of $\lambda {{\mathbb {B}}}_G(x)$.

2.1 Uniform Metric Spaces and Quasi-hyperbolic Metrics

A curve in (X, d) is a continuous function $\gamma :\; I\rightarrow X$ from an interval $I\subset {{\mathbb {R}}}$ to X. We use $\gamma$ to denote both the function and its image set. The length $\ell (\gamma )$ of $\gamma$ with respect to the metric d is defined in the usual way. The parameter interval I is allowed to be closed, open or half-open. If $\ell (\gamma ) < \infty$, then $\gamma$ is said to be rectifiable. Obviously, when I is compact, $\gamma$ is a continuum (i.e., a connected and compact set) in X since both the connectedness and the compactness are invariant under continuous functions.

A metric space (X, d) is called geodesic if every pair of points x and $y\in X$ can be joined by a curve $\gamma$ such that

$$\begin{aligned} \ell (\gamma )=|x-y|, \end{aligned}$$

and the curve $\gamma$ is also called a geodesic in X joining x and y.

A metric space (X, d) is called a-uniform if there is a constant $a\ge 1$ such that each pair of points x and y in X can be joined by a rectifiable curve $\gamma$ in X satisfying

• (1) $\min \{\ell (\gamma [x,z]),\ell (\gamma [z,y])\}\le a\,d_X(z)$ for all $z\in \gamma$, and

• (2) $\ell (\gamma )\le a\,|x-y|$.

If the metric d is replaced by its inner metric $\sigma _X$, then uniformity is called inner uniformity, where the inner metric $\sigma _X$ is defined by

$$\begin{aligned} \sigma _X(x,y)=\inf _{\gamma \in \Gamma (x,y: X)}\big \{\ell (\gamma )\big \} \end{aligned}$$

for any x and y in X. Here and in what follows, $\Gamma (x,y: X)$ denotes the family of all rectifiable curves in X connecting x and y.

When our uniform space (resp. our inner uniform space) is a domain of a metric space, we call it a uniform domain (resp. an inner uniform domain).

Assume that (X, d) is a rectifiably connected, non-complete and locally compact metric space. The quasi-hyperbolic metric $k_X$ in X (with respect to d) is defined by

$$\begin{aligned} k_X(x, y)=\inf \Big \{\int _{\gamma } \frac{1}{d_X(z)}\;|dz|\Big \}, \end{aligned}$$

where the infimum is taken over all curves in $\Gamma (x,y: X)$, $d_X(z)$ denotes the distance from z to $\partial X$, and |dz| denotes the arc-length element with respect to d. The quantity $\int _{\gamma } \frac{1}{d_X(z)}\;|dz|=\ell _{k,X}(\gamma )$ is called the quasi-hyperbolic length of $\gamma$. A curve in X connecting x and y is called a quasihyperbolic geodesic if $\ell _{k,X}(\gamma )=k_X(x,y)$.

For x and $y\in X$, their relative distance is defined by the number:

$$\begin{aligned} r_X(x,y)=\frac{|x-y|}{\min \{d_X(x),d_X(y)\}}. \end{aligned}$$

The following lemma is useful.

Lemma 2.1

Suppose that (X, d) is a noncomplete and c-quasiconvex metric space, and let $\tau > 0$ be a constant. For any pair of points x, y in X,

• 1 if $|x-y|\le \tau \max \{d_X(x), d_X(y)\}/c(\tau +1)$, then

  $$\begin{aligned} k_X(x,y)\le c(\tau +1)\frac{|x-y|}{\max \{d_X(x), d_X(y)\}} \le c(\tau +1)r_X(x,y); \end{aligned}$$

• 2 if $k_X(x,y)\le \tau$, then

  $$\begin{aligned} k_X(x,y)\le c(\tau +1)r_X(x,y); \end{aligned}$$

• 3 if $k_X(x,y)\ge \tau$, then

  $$\begin{aligned} |x-y|\ge \frac{\tau }{c(\tau +1)}\max \{d_X(x), d_X(y)\}; \end{aligned}$$

Proof

(1) Assume that $|x-y|\le \tau \max \{d_X(x), d_X(y)\}/c(\tau +1)$. Without loss of generality, assume that $\max \{d_X(x), d_X(y)\}=d_X(x)$. Since X is c-quasiconvex, we know that there is a curve $\gamma _{xy}$ in X connecting x and y such that

$$\begin{aligned} \ell (\gamma _{xy})\le c|x-y|, \end{aligned}$$

and so,

$$\begin{aligned} \ell (\gamma _{xy})\le \frac{\tau }{\tau +1}d_X(x). \end{aligned}$$

Since for every $z\in \gamma _{xy}$,

$$\begin{aligned} d_X(z)\ge d_X(x)-\ell (\gamma _{xy})\ge \frac{1}{\tau +1}d_X(x), \end{aligned}$$

and since

$$\begin{aligned} k_X(x,y)\le \int _{\gamma _{xy}}\frac{|dz|}{d_X(z)}, \end{aligned}$$

we get

$$\begin{aligned} k_X(x,y)\le c(\tau +1)\frac{|x-y|}{d_X(x)} \le c(\tau +1)r_X(x,y). \end{aligned}$$

(2) Assume that $k_X(x,y)\le \tau$. By the statement (1) of the lemma, we may assume that $|x-y|\ge \tau \max \{d_X(x), d_X(y)\}/c(\tau +1)$. Then we deduce that

$$\begin{aligned} k_X(x,y)\le \tau \le c(\tau +1)\frac{|x-y|}{\max \{d_X(x), d_X(y)\}}\le c(\tau +1)r_X(x,y). \end{aligned}$$

(3) Assume that $k_X(x,y)\ge \tau$. Without loss of generality, we assume that $\max \{d_X(x), d_X(y)\}=d_X(x)$. Suppose on the contrary that

$$\begin{aligned} |x-y|< \tau d_X(x)/c(\tau +1). \end{aligned}$$

Then by the statement (1) of the lemma, we get

$$\begin{aligned} k_X(x,y)\le c(\tau +1)\frac{|x-y|}{d_X(x)}< \tau . \end{aligned}$$

This obvious contradiction proves this statement. $\square$

For two nondegenerate and bounded sets E and F in X, let

$$\begin{aligned} \Delta (E,F)=\frac{{{\text {dist}}}(E,F)}{\min \{{{\text {diam}}}E, {{\text {diam}}}F\}}, \end{aligned}$$

which is called the relative distance between E and F.

Based on Lemma 2.1, we have the following sufficient condition for two Whitney balls to be disjoint.

Lemma 2.2

Suppose that X is a metric space and $G\subset X$ is a c-quasiconvex domain. Let $\alpha >1/2$ and $0<\lambda \le 1/4(\alpha +2)c$. For x, $y\in G$, if $k_G(x,y)\ge \alpha$, then

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\ge 2\alpha -1\;\;\text{ and }\;\;\lambda {{\overline{{{\mathbb {B}}}}}}_G(x) \cap \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset . \end{aligned}$$

Proof

Let x and y be two points in G with $k_G(x,y)\ge \alpha$. Then it follows from Lemma 2.1(3) that

$$\begin{aligned} |x-y|> \frac{\alpha }{c(\alpha +2)}\max \{d_G(x), d_G(y)\}. \end{aligned}$$

Obviously, $\lambda {{\overline{{{\mathbb {B}}}}}}_G(x)\subset \lambda {{\overline{{{\mathbb {B}}}}}}(x)$ and $\lambda {{\overline{{{\mathbb {B}}}}}}_G(y)\subset \lambda {{\overline{{{\mathbb {B}}}}}}(y)$, and thus, we have

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y)) \ge \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}(x),\lambda {{\overline{{{\mathbb {B}}}}}}(y)). \end{aligned}$$

Since

$$\begin{aligned} {{\text {dist}}}(\lambda {{\overline{{{\mathbb {B}}}}}}(x),\lambda {{\overline{{{\mathbb {B}}}}}}(y)) \ge |x-y|-\lambda d_G(x)-\lambda d_G(y) \end{aligned}$$

and

$$\begin{aligned} {{\text {diam}}}(\lambda {{\overline{{{\mathbb {B}}}}}}(x))\le 2\lambda d_G(x), \end{aligned}$$

we know that

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\ge 2\alpha -1>0, \end{aligned}$$

since $\alpha >1/2$. Also, this implies that

$$\begin{aligned} \lambda {{\overline{{{\mathbb {B}}}}}}_G(x)\cap \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset . \end{aligned}$$

The proof of the lemma is complete. $\square$

Remark 2.1

For any pair of points x, $y\in G\varsubsetneq {{\mathbb {R}}}^n$, if $k_G(x,y)\ge \alpha >2\log 2$, then

$$\begin{aligned} {{\mathbb {B}}}_{k_{G}}\left( x, \frac{\alpha }{2}\right) \cap {{\mathbb {B}}}_{k_{G}} \left( y, \frac{\alpha }{2}\right) =\emptyset , \end{aligned}$$

where ${{\mathbb {B}}}_{k_{G}}\left( x, \frac{\alpha }{2}\right) =\{z\in G:\; k_{G}(z,x)<\frac{\alpha }{2}\}$, the quasihyperbolic ball with center x and radius $\frac{\alpha }{2}$.

Since for any $z\in G$, $\left( 1-e^{-\frac{\alpha }{2}}\right) {{\mathbb {B}}}(z)\subset {{\mathbb {B}}}_{k_{G}}\left( z, \frac{\alpha }{2}\right)$ (cf. [10, (5.15)]), we get

$$\begin{aligned} \left( 1-e^{-\frac{\alpha }{2}}\right) {{\mathbb {B}}}(x)\cap \left( 1-e^{-\frac{\alpha }{2}}\right) {{\mathbb {B}}}(y)=\emptyset . \end{aligned}$$

Obviously,

$$\begin{aligned} \frac{1}{2}{{\overline{{{\mathbb {B}}}}}}(x)\cap \frac{1}{2}{{\overline{{{\mathbb {B}}}}}}(y)=\emptyset , \end{aligned}$$

since $1-e^{-\frac{\alpha }{2}}>\frac{1}{2}$.

A minimally nice (i.e., rectifiably connected, non-complete and locally complete) metric space (X, d) is called quasihyperbolic $\varphi$-uniform (briefly, QH $\varphi$-uniform) if there is a self-homeomorphism $\varphi$ of $[0,+\infty )$ such that for any pair of points x and $y\in X$,

$$\begin{aligned} k_X(x,y)\le \varphi (r_X(x,y)). \end{aligned}$$

When the QH $\varphi$-uniform space is a domain of a metric space, it is called a QH $\varphi$-uniform domain.

We remark that local compactness implies local completeness. Also, it is known that uniformity is quantitatively equivalent to QH $\varphi$-uniformity with $\varphi$ being slow, i.e., $\limsup _{t\rightarrow \infty }\varphi (t)/t<1$, in Banach spaces; see [21, Theorem 6.16]. For general metric spaces, Buckley and Herron established the following generalization.

Theorem 2

[7, Theorem 3.1]. A minimally nice and locally $(c,\lambda )$-quasiconvex space (X, d) is uniform if and only if it is QH $\varphi$-uniform with a slow $\varphi$. The uniformity constant depends only on c and $\varphi$, and conversely in an a-uniform space, one can always take $\varphi (t)=4a^2\log (1+t)$.

2.2 Modulus and Capacity

Let $(X,d,\mu )$ denote a metric measure space, and let $\Gamma$ (sometimes with a subscript) be a family of curves in X. For a number $1\le p< \infty$, the p-modulus of $\Gamma$ is defined as

$$\begin{aligned} \textrm{mod}_p \Gamma =\inf \Big \{\int _{X}\varrho ^p d\mu \Big \}, \end{aligned}$$

where the infimum is taken over all nonnegative Borel functions $\varrho : X\rightarrow [0,\infty ]$ satisfying

$$\begin{aligned} \int _{\gamma } \varrho ds\ge 1 \end{aligned}$$

(2.1)

for all locally rectifiable curves $\gamma \in \Gamma$. Functions $\varrho$ satisfying (2.1) are called admissible densities for $\Gamma$.

The moduli of families of curves possess the following elementary and useful properties.

Theorem 3

[12, §2.3]. For $1\le p< \infty$, the following statements are true:

• 1 If $\Gamma =\emptyset$, the empty set, then $\textrm{mod}_p(\Gamma )=0$;

• 2 For two families of curves $\Gamma _1$ and $\Gamma _2$ in X, if $\Gamma _1\subset \Gamma _2$, then

  $$\begin{aligned} \textrm{mod}_p \Gamma _1\le \textrm{mod}_p \Gamma _2; \end{aligned}$$

• 3 For two families of curves $\Gamma _3$ and $\Gamma _4$ in X, if each curve $\gamma _4\in \Gamma _4$ has a subcurve $\gamma _3$ which belongs to $\Gamma _3$, then

  $$\begin{aligned} \textrm{mod}_p \Gamma _4\le \textrm{mod}_p \Gamma _3; \end{aligned}$$

• 4 For a sequence of families of curves $\{\Gamma _i\}_{i=1}^{+\infty }$ in X,

  $$\begin{aligned} \textrm{mod}_p \left( \bigcup \limits _{i=1}^{+\infty }\Gamma _i\right) \le \sum _{i=1}^{+\infty }\textrm{mod}_p \Gamma _i. \end{aligned}$$

Let G be an open set in X, and let u be an arbitrary real-valued function in G. We say that a Borel function $\rho : G\rightarrow [0,\infty ]$ is a very weak gradient of u in G if for every pair of points x and $y\in G$,

$$\begin{aligned} |u(x)-u(y)|\le \int _{\gamma }\rho ds \end{aligned}$$

whenever $\gamma$ belongs to $\Gamma (x,y:G)$, the set of all rectifiable curves in G connecting x and y.

For an open subset G of X, and its two disjoint closed subsets E and F, let the triple (E, F; G), which is called a condenser, denote the family of all curves in G joining E and F. With the aid of very weak gradients, Heinonen and Koskela gave the following definition of capacity in the metric space setting.

Definition 2.1

For a condenser (E, F; G) of X, its p-capacity for $1\le p<\infty$ is defined as

$$\begin{aligned} \textrm{cap}_p(E,F;G)=\inf \Big \{\int _{G}\rho ^p d\mu \Big \}, \end{aligned}$$

where the infimum is taken over all very weak gradients $\rho$ of each admissible function u for the condenser (E, F; G). Here, a function u in G is called admissible for a condenser (E, F; G) if the restriction $u|_{E}\ge 1$ and the restriction $u|_{F}\le 0$.

Heinonen and Koskela showed that for all $1\le p<\infty$,

$$\begin{aligned} \textrm{cap}_p(E,F;G)=\textrm{mod}_p(E,F;G) \end{aligned}$$

(see [12, Proposition 2.17]). Thus, in the following, we use the notation $\textrm{cap}_p(E,F;G)$ instead of $\textrm{mod}_p(E,F;G)$.

2.3 Capacity Conditions and Regularity

In order to generalize the theory of quasiconformal mappings to the metric space setting, in [12], Heinonen and Koskela introduced the concept of a Loewner space. The definition is as follows.

Definition 2.2

Suppose that $(X,d,\mu )$ is a rectifiably connected metric measure space. It is called

• (1) a Q-Loewner space if there is a function $\phi :(0,+\infty )\rightarrow (0,+\infty )$ so that

  $$\begin{aligned} \textrm{cap}_Q(E,F;X)\ge \phi (t) \end{aligned}$$

  (2.2)

  whenever E and F are two disjoint and nondegenerate continua in X, and

  $$\begin{aligned} t\ge \Delta (E,F), \end{aligned}$$

  where $\phi$ is called a Loenwer control function of X. Moreover, if a domain $G\subset X$ satisfies the Q-Loewner condition (2.2), then we say that G is a Q-Loewner domain.

• (2) a locally Q-Loewner space if X is locally compact and there exist numbers $\kappa \ge 1$, $\varepsilon _0\in (0,\kappa ^{-1}]$, and a function $\phi :(0,+\infty )\rightarrow (0,+\infty )$ so that

  $$\begin{aligned} \textrm{cap}_Q(E,F;{{\mathbb {B}}}(x,\varepsilon \kappa d_G(x)))\ge \phi (t) \end{aligned}$$

  (2.3)

  whenever $x\in X$, $0<\varepsilon <\varepsilon _0$, E and $F\subset {{\mathbb {B}}}(x,\varepsilon d_G(x))$ are two disjoint and nondegenerate continua, and $t\ge \Delta (E,F)$.

By [12, Theorem 3.13], every Q-regular Loewner space (see Definition 2.3 for the definition of Q-regularity) satisfies the following properties.

Quasiconvexity A metric space (X, d) is said to be c-quasiconvex if there is a constant $c\ge 1$ such that for each pair of points x and $y\in X$, there exists a curve $\gamma$ joining x and y satisfying

$$\begin{aligned} \ell (\gamma )\le c|x-y|. \end{aligned}$$

Such a curve is also called a c-quasiconvex curve.

Local quasiconvexity A metric space (X, d) is said to be locally $(c,\alpha )$-quasiconvex if there are constants $c\ge 1$ and $0<\alpha <1$ such that for each point $w\in X$, every pair of points in $\alpha B(w)$ can be joined by a c-quasiconvex curve.

Linearly local connectedness A metric space (X, d) is said to be b-linearly locally connected (briefly, b-LLC) if there is a constant $b\ge 1$ so that for each $x\in X$ and $r>0$, the following conditions hold:

LLC$_1:$:

  any pair of points in ${{\mathbb {B}}}(x,r)$ can be joined by a rectifiable curve in ${{\mathbb {B}}}(x,br)$;

LLC$_2:$:

  any pair of points in $X\setminus {{\overline{{{\mathbb {B}}}}}}(x,r)$ can be joined by a rectifiable curve in $X\setminus {{\overline{{{\mathbb {B}}}}}}(x,r/b)$.

Locally external connectedness If a metric space (X, d) is locally compact and the LLC$_2$ condition holds for all $x\in X$ and all $r\in (0,d_X(x)/b)$ with $b>1$, then we say that X is b-locally externally connected (briefly, b-LEC).

Next is the definition of (Ahlfors-David) regularity.

Definition 2.3

Suppose that $(X,d,\mu )$ is a metric measure space. If there is a constant $C_r\ge 1$ such that

$$\begin{aligned} C_r^{-1}R^Q\le \mu ({{\mathbb {B}}}(x,R))\le C_rR^Q \end{aligned}$$

for all balls ${{\mathbb {B}}}(x,R)$ in X of radius $R< {{\text {diam}}}X$, then X is said to be Q-regular. In particular, when only the upper bound above holds, X is called upper Q-regular, and when only the lower bound above holds, X is called lower Q-regular.

We make a notational convention: In the following, $C_r$ always denotes the constant from the definition of regularity, and also, it is called a regularity constant of X. Its value may be different at different occasions.

It is known that every Q-regular metric measure space with $Q>1$ has Hausdorff dimension precisely Q (cf. [11, §8.7]). The converse is invalid. This can be seen from the first statement of Example 5.1 or Example 5.2 below.

In [12], Heinonen and Koskela discussed the relationship between regularity and capacity; see [12, Lemma 3.14]. In [12, Lemma 3.14], the underlying spaces are assumed to be Loewner. But this assumption is not used in its proof. Thus we formulate it in the following form which we need.

Theorem 4

[12, Lemma 3.14] Suppose that $(X,d,\mu )$ is Q-regular with $Q>1$. Then there is a constant $C_1>0$ such that for any r with $0<2r<R<{{\text {diam}}}X$ and $y\in X$,

$$\begin{aligned} \textrm{cap}_Q({{\overline{{{\mathbb {B}}}}}}(y,r),X\setminus {{\mathbb {B}}}(y,R); X) \le C_1\left( \log \frac{R}{r}\right) ^{1-Q}. \end{aligned}$$

The following result is from [11].

Theorem 5

[11, Proposition 8.19 and Theorem 8.23]. Let $(X,d,\mu )$ be a Q-Loewner space. Then there is a constant $C_2\ge 1$ such that

$$\begin{aligned} C_2^{-1}R^Q\le \mu ({{\mathbb {B}}}(x,R)) \end{aligned}$$

(2.4)

for all balls ${{\mathbb {B}}}(x,R)$ in X of radius $R<{{\text {diam}}}X$. If X is upper Q-regular, then $(X,d,\mu )$ is Q-regular and there is a decreasing self-homeomorphism $\psi$ of $(0,\infty )$ such that

$$\begin{aligned} \textrm{cap}_Q(E,F;X)\ge \psi (\Delta (E,F)) \end{aligned}$$

(2.5)

for all disjoint and nondegenerate continua E and F in X. Moreover, we can select $\psi$ so as to satisfy

$$\begin{aligned} \psi (t)\approx \log \frac{1}{t} \end{aligned}$$

for all sufficiently small t, and

$$\begin{aligned} \psi (t)\approx (\log t)^{1-Q} \end{aligned}$$

(2.6)

for all sufficiently large t. The statement is quantitative.

It is known that all Euclidean spaces ${{\mathbb {R}}}^n$ are 1-quasiconvex. In combination with Lemma 2.2, Remark 2.1 and the definitions of the k-cap conditions in ${{\mathbb {R}}}^n$ (cf. [5, 6]), we introduce the following definition in the quasiconvex metric space setting.

Definition 2.4

Suppose that $(X,d,\mu )$ is a metric measure space, and $G\subset X$ is a locally compact and c-quasiconvex domain. Let $\tau >0$ and $0<\lambda \le 1/(16c)$ be constants. Then G is said to satisfy a $(\tau ,\lambda )$-k-cap condition if for every pair of points x and $y\in G$ with $k_G(x,y)\ge 2$,

$$\begin{aligned} \textrm{cap}_Q\big (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y);G\big )\le \frac{\tau }{k_G(x,y)^{Q-1}}. \end{aligned}$$

(2.7)

Also, G is called a $(\tau ,\lambda )$-k-cap domain. When the values of the parameters are not important, briefly, we say that G is a k-cap domain.

Remark 2.2

By [12], Heinonen and Koskela proved that the combination of Loewner property and regularity guarantees quasiconvexity (see [12, Theorem 3.13] or Lemma 3.5 below). Based on this, it follows from the proof of Theorem 1.1 that the assumption of G being quasiconvex can be removed from Theorem 1.1. As we see from Definition 2.4 that the concept of k-cap condition is introduced in quasiconvex metric measure spaces or their domains, such an assumption is kept in Theorem 1.1.

2.4 Weak Slice Conditions

In order to study the geometrical characterizations of domains which support Sobolev–Poincaré-type imbeddings, Buckley and Koskela introduced a concept which is the so-called slice condition (cf. [8]). Later, in order to establish sharp inequalities of Trudinger-type on general domains, Buckley and O’Shea generalized the slice condition into a weaker one which is called the weak slice condition (cf. [9]). See [7] for an application of this condition. The following is the definition of weak slice conditions.

Definition 2.5

Suppose that (X, d) is a b-LEC metric space, x and $y\in X$ are two points.

• (1) A non-empty bounded open subset $S\subset X$ is an A-slice separating x and y if there is a constant $A\ge 2$ such that

  $$\begin{aligned} A^{-1}{{\mathbb {B}}}(x)\cap S=\emptyset =S\cap A^{-1}{{\mathbb {B}}}(y), \end{aligned}$$

  and for all $\gamma \in \Gamma (x,y: X)$,

  $$\begin{aligned} \ell (\gamma \cap S)\ge A^{-1}{{\text{ diam }}}(S); \end{aligned}$$

• (2) For x and y in X, let ${\mathcal {S}}(x,y)$ denote a collection of pairwise disjoint A-slices separating x and y. Define

  $$\begin{aligned} d_{ws}(x,y)=d_{ws}(x,y;A)=1+\sup \big \{ \textrm{card} {\mathcal {S}}(x,y)\big \}, \end{aligned}$$

  where the supremum is taken over all ${\mathcal {S}}(x,y)$ and card means cardinality. We say that X satisfies a weak A-slice condition if for each pair of points x and y in X,

  $$\begin{aligned} k_X(x,y)\le Ad_{ws}(x,y). \end{aligned}$$

  (2.8)

In [7], Buckley and Herron considered the characterization of uniform spaces in terms of the weak slice condition. The obtained result is as follows.

Theorem 6

[7, Theorem 4.2]. A minimally nice metric space (X, d) is uniform and LEC if and only if it is quasiconvex, LLC and satisfies a weak slice condition. These implications are quantitative.

3 Auxiliary Results

The aim of this section is to establish several auxiliary lemmas, which are useful for the proof of Theorem 1.1. The first lemma compares the relative distance between the closures of two Whitney balls with the one between their centers.

Lemma 3.1

Suppose that $G\subset X$ is a c-quasiconvex domain, where the case $G=X$ is included when (X, d) is noncomplete. For any pair of points x,  y in G and $0< \lambda \le 1/(8c)$, if $\lambda {{\overline{{{\mathbb {B}}}}}}_G(x)\cap \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset$ and $k_G(x,y)> 1$, then

$$\begin{aligned} \frac{1}{4\lambda }r_G(x,y)\le \Delta (\lambda \overline{{{\mathbb {B}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\le \frac{1}{\lambda }r_G(x,y). \end{aligned}$$

(3.1)

Proof

We only need to prove the case $G\subsetneq X$ since the proof of the case $G=X$ is similar. Without loss of generality, we may assume that $d_G(x)\le d_G(y)$. Since G is c-quasiconvex and open, we infer from [15, Lemma 3.1] that

$$\begin{aligned} \partial \lambda {{\mathbb {B}}}_G(x)\ne \emptyset , \end{aligned}$$

and from [15, Lemma 3.5] that for any $u\in \tau {{\overline{{{\mathbb {B}}}}}}_G(z)$,

$$\begin{aligned} |u-z|=\tau d_G(z)\;\;\text{ if } \text{ and } \text{ only } \text{ if }\;\;u\in \partial \tau {{\mathbb {B}}}_G(z). \end{aligned}$$

These imply that there is a point $z\in \partial \lambda {{\mathbb {B}}}_G(x)$ such that

$$\begin{aligned} {{\text {diam}}}\lambda {{\overline{{{\mathbb {B}}}}}}_G(x) \ge |x-z|= \lambda d_G(x). \end{aligned}$$

Since $|x-y|\ge {{\text {dist}}}(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda \overline{{{\mathbb {B}}}}_G(y))$, we obtain that

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\le \frac{|x-y|}{\lambda d_G(x)}=\frac{1}{\lambda }r_G(x,y). \end{aligned}$$

This proves the right side of (3.1).

Since $k_G(x,y)> 1$, it follows from Lemma 2.1() that

$$\begin{aligned} |x-y|\ge \frac{1}{2c}d_G(y). \end{aligned}$$

Then the assumption $0< \lambda \le 1/(8c)$ implies that

$$\begin{aligned} {{\text {dist}}}(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\ge |x-y|-\lambda d_G(x)-\lambda d_G(y)\ge \frac{1}{2}|x-y|, \end{aligned}$$

and thus, we get

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\ge \frac{|x-y|}{4\lambda d_G(x)}=\frac{1}{4\lambda }r_G(x,y), \end{aligned}$$

since $\min \{{{\text {diam}}}\lambda {{\overline{{{\mathbb {B}}}}}}_G(x), {{\text {diam}}}\lambda \overline{{{\mathbb {B}}}}_G(x)\}\le 2\lambda d_G(x)$. This proves the left side of (3.1), and hence the lemma is proved. $\square$

It is known that for any domain $G\subset {{\mathbb {R}}}^n$, and for any $\lambda \in (0, 1)$ and $x\in G$, the closure $\lambda {{\overline{{{\mathbb {B}}}}}}_G(x)$ of the Whitney ball $\lambda {{\mathbb {B}}}_G(x)=\lambda {{\mathbb {B}}}(x)$ is a continuum. But the fact is invalid in the metric space setting. The next lemma establishes the existence of continua in such closures of Whitney balls, which also satisfy certain comparability conditions.

Lemma 3.2

Suppose that $G\subset X$ is a c-quasiconvex domain, where the case $G=X$ is included when (X, d) is noncomplete. For any pair of points x, y in G and $0< \lambda \le 1/(8c)$, if $\lambda {{\overline{{{\mathbb {B}}}}}}_G(x)\cap \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset$ and $k_G(x,y)>1$, then there are curves $\gamma _x\subset \lambda {{\overline{{{\mathbb {B}}}}}}_G(x)$ and $\gamma _y\subset \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)$ such that both $\gamma _x$ and $\gamma _y$ are continua, and

$$\begin{aligned} \frac{1}{4\lambda }r_G(x,y)\le \Delta (\gamma _x,\gamma _y)\le \frac{3c}{2\lambda }r_G(x,y). \end{aligned}$$

(3.2)

Proof

We only need to prove the case $G\subsetneq X$ since the proof of the case $G=X$ is similar. Since G is c-quasiconvex and open, we know from [15, Lemmas 3.1 and 3.5] that there are points $x_1\in \partial (\lambda /c){{\mathbb {B}}}_G(x)$ and $y_1\in \partial (\lambda /c) {{\mathbb {B}}}_G(y)$ such that

$$\begin{aligned} |x_1-x|=\frac{1}{c}\lambda d_G(x)\;\;\text{ and } \;\;|y_1-y|=\frac{1}{c}\lambda d_G(y). \end{aligned}$$

Also, we know that there are a c-quasiconvex curve $\gamma _x$ in G connecting $x_1$ and x, and another c-quasiconvex curve $\gamma _y$ in G connecting $y_1$ and y. Then $\gamma _x\subset \lambda {\overline{{{\mathbb {B}}}}}_G(x)$ and $\gamma _y\subset \lambda {\overline{{{\mathbb {B}}}}}_G(y)$, and both of them are continua. Since

$$\begin{aligned} {{\text {dist}}}(\gamma _x,\gamma _y)\ge {{\text {dist}}}(\lambda {\overline{{{\mathbb {B}}}}}_G(x), \lambda {\overline{{{\mathbb {B}}}}}_G(y))\;\;\text{ and }\;\; \frac{\lambda }{c} d_G(x)\le {{\text {diam}}}(\gamma _x)\le 2\lambda d_G(x), \end{aligned}$$

it follows from Lemma 3.1 that

$$\begin{aligned} \Delta (\gamma _x,\gamma _y)\ge \Delta (\lambda {\overline{{{\mathbb {B}}}}}_G(x),\lambda {\overline{{{\mathbb {B}}}}}_G(y))\ge \frac{1}{4\lambda }r_G(x,y). \end{aligned}$$

Next, we check the upper bound in (3.2). Since $k_G(x,y)>1$, by Lemma 2.1(), we have that

$$\begin{aligned} |x-y|\ge \frac{1}{2c}d_G(y). \end{aligned}$$

This implies that

$$\begin{aligned} {{\text {dist}}}(\gamma _x,\gamma _y)\le |x-y|+2\lambda d_G(y)\le & {} (1+4c\lambda )|x-y|\le \frac{3}{2}|x-y|. \end{aligned}$$

Hence we obtain that

$$\begin{aligned} \Delta (\gamma _x,\gamma _y)\le \frac{c}{\lambda }\frac{{{\text {dist}}}(\gamma _x,\gamma _y)}{\min \{d_G(x), d_G(y)\}}\le \frac{3c}{2\lambda }r_G(x,y), \end{aligned}$$

which is what we need. $\square$

The following lemma is an analog of [6, Lemma 2.2] in the Q-regular metric measure space setting.

Lemma 3.3

Suppose that $(X,d,\mu )$ is a Q-regular metric measure space with $Q>1$. Let E, F be disjoint bounded closed sets in X. Then the following two statements are true:

• 1 $\text {cap}_Q(E,F;X)\le C_r(1+1/\Delta (E,F))^Q;$ (Recall that the constant $C_r$ denotes a regularity constant of X.)

• 2 If $\Delta (E,F)\ge 2$, then

  $$\begin{aligned} \textrm{cap}_Q(E,F;X)\le C_1\left( \log \Delta (E,F)\right) ^{1-Q}, \end{aligned}$$

  where the constant $C_1$ is from Theorem 4.

Proof

Let $a={{\text {diam}}}(E)$ and $b={{\text {dist}}}(E,F)$. Without loss of generality, we assume that ${{\text {diam}}}(E)\le {{\text {diam}}}(F)$.

(1) For $x\in E$, let

$$\begin{aligned} \varrho (y)={\left\{ \begin{array}{ll} \displaystyle \; 1/b, &{} \text{ if }\quad y\in {{\mathbb {B}}}(x,a+b),\\ \;\;\;\ 0, &{} \;\text{ otherwise }. \end{array}\right. } \end{aligned}$$

Then $\varrho$ is admissible for the curve family $\Gamma (E,F:X)$. Since $(X,d,\mu )$ is Q-regular, we deduce that

$$\begin{aligned} \textrm{cap}_Q(E,F;X) \le \int _{X}\varrho ^Q d\mu \le \mu ({{\mathbb {B}}}(x,a+b))/b^Q \le C_r(1+1/\Delta (F,F))^Q, \end{aligned}$$

where $C_r\ge 1$ is a regularity constant of X.

(2). The assumption $\Delta (E,F)\ge 2$ implies that $b\ge 2a$. Then for $x\in E$, we see that $E\subset {{\mathbb {B}}}(x,a)$ and $F\subset X\setminus {{\mathbb {B}}}(x,b)$. By Theorems 3(3) and 4, we have that

$$\begin{aligned} \textrm{cap}_Q(E,F;X)\le \textrm{cap}_Q({{\overline{{{\mathbb {B}}}}}}(y,a),X\setminus {{\mathbb {B}}}(y,b);X)\le C_1\left( \log \Delta (E,F)\right) ^{1-Q}, \end{aligned}$$

where the constant $C_1$ is from Theorem 4. $\square$

Lemma 3.4

Suppose that $(X,d,\mu )$ is a Q-regular metric measure space and $G\subset X$ is a domain. If G is Q-Loewner, then G is Q-regular.

Proof

Let $x\in G$ and $0<r<{{\text {diam}}}G$. Then the inequality (2.4) in Theorem 5 ensures that there is a constant $C_2 \ge 1$ such that

$$\begin{aligned} \mu ({{\mathbb {B}}}_G(x,r))\ge C^{-1}_2r^Q. \end{aligned}$$

Moreover, since X is Q-regular, we see that

$$\begin{aligned} \mu ({{\mathbb {B}}}_G(x,r))\le \mu ({{\mathbb {B}}}(x,r))\le C_rr^Q. \end{aligned}$$

Let $C_3=\max \{C_r,C_2\}$. Then G is Q-regular with a regularity constant $C_3$. $\square$

Lemma 3.5

Suppose that $(X,d,\mu )$ is a Q-regular metric measure space and $G\subset X$ is a domain. If G is Q-Loewner, then G is c-quasiconvex and c-LLC.

Proof

It follows from Lemma 3.4 that G is Q-regular, and thus, the lemma follows from [12, Theorem 3.13]. $\square$

4 Proof of Theorem 1.1

The aim of this section is to prove Theorem 1.1. We shall prove the theorem by checking the implications: $({\mathfrak {a}})\Rightarrow ({\mathfrak {b}})\Rightarrow ({\mathfrak {c}})\Rightarrow ({\mathfrak {d}})\Rightarrow ({\mathfrak {a}})$. We only need to prove the case when G is a proper domain in X since the proof of the case when $G=X$ is similar.

$({\mathfrak {a}})\Rightarrow ({\mathfrak {b}})$. Assume that G is a-uniform and locally Q-Loewner. This implication follows from Theorem A since uniformity implies inner uniformity.

$({\mathfrak {b}})\Rightarrow ({\mathfrak {c}})$. Assume that G is Q-Loewner and a-inner uniform. Since G is c-quasiconvex, it follows that for any pair of points x, $y\in G$, there is a curve $\gamma _{xy}$ in G connecting x and y such that

$$\begin{aligned} \sigma _G(x,y)\le l(\gamma _{xy})\le c|x-y|. \end{aligned}$$

Moreover, the assumption of G being a-inner uniform and Theorem 2 guarantee that

$$\begin{aligned} k_G(x,y)\le 4a^2\log \left( 1+\frac{\sigma _G(x,y)}{\min \{d_G(x),d_G(y)\}}\right) . \end{aligned}$$

Then it follows that

$$\begin{aligned} k_G(x,y)\le 4a^2\log \left( 1+cr_G(x,y)\right) . \end{aligned}$$

By letting $\varphi (t)=4a^2\log (1+ct)$, we see that G is QH $\varphi$-uniform with $\varphi$ being slow. Again, we know from Theorem 2 that G is $a_{1}$-uniform. Since Lemma 3.5 indicates G is c-LLC, and thus, it is c-LEC. Hence we know from Theorem 6 that G satisfies a weak A-slice condition, which completes the proof of this implication.

$({\mathfrak {c}})\Rightarrow ({\mathfrak {d}})$. Assume that G is Q-Loewner and satisfies a weak A-slice condition. Let x, y be two points in G with $k_G(x,y)\ge 2$. To prove this implication, it suffices to show that there are constants $\tau >0$ and $0<\lambda \le 1/(16c)$, independent of x and y, such that

$$\begin{aligned} \textrm{cap}_Q\Big (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y);G\Big )\le \frac{\tau }{k_G(x,y)^{Q-1}}. \end{aligned}$$

(4.1)

Since G is Loewner, Lemma 3.5 ensures that G is c-LLC, and then, we know from the assumptions of G being c-quasiconvex and satisfying a weak slice condition, together with Theorem 6, that it is a-uniform. Without loss of generality, we assume that $a=c$ and $d_G(x)\le d_G(y)$. Let

$$\begin{aligned} \lambda =\min \Big \{\frac{1}{16c}, \frac{1}{4}(1-e^{-1/(2c^2)})\Big \}. \end{aligned}$$

(4.2)

Then the assumption $k_G(x,y)\ge 2$ and Lemma 2.2 imply that

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\ge 3 \;\;\text{ and }\;\; \lambda {{\overline{{{\mathbb {B}}}}}}_G(x)\cap \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset . \end{aligned}$$

(4.3)

By the assumptions that X is Q-regular and G is Loewner, we know from Lemma 3.4 that G is Q-regular. Then Lemma 3.3(2) ensures that there is a constant $C_1\ge 1$ such that

$$\begin{aligned} \textrm{cap}_Q(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y);G)\le C_1\left( \log \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x), \lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\right) ^{1-Q}. \end{aligned}$$

(4.4)

Moreover, by the assumption $k_G(x,y)\ge 2$ and Lemma 2.1(), we have that $r_G(x,y)\ge 2/(3c)$. Then we deduce from (4.2) that

$$\begin{aligned} \frac{1}{4\lambda }r_G(x,y)\ge \frac{8}{3}. \end{aligned}$$

(4.5)

Since the assumption $k_G(x,y)\ge 2$, (4.3) and Lemma 3.1 guarantee that

$$\begin{aligned} \Delta (\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y))\ge \frac{1}{4\lambda }r_G(x,y), \end{aligned}$$

we infer from (4.4) and (4.5) that

$$\begin{aligned} \textrm{cap}_Q(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y);G)\le C_1\left( \log \frac{1}{4\lambda }r_G(x,y)\right) ^{1-Q}. \end{aligned}$$

(4.6)

We still need the following relation between $r_G(x,y)$ and $k_G(x,y)$:

$$\begin{aligned} r_G(x,y)\ge e^{k_G(x,y)/4c^2}-1. \end{aligned}$$

(4.7)

This estimate directly follows from Theorem 2 since G is a-uniform.

Since it follows from (4.2), together with the assumption $k_G(x,y)\ge 2$, that

$$\begin{aligned} \frac{1}{4\lambda }\Big (e^{k_G(x,y)/4c^2}-1\Big )>1, \end{aligned}$$

by substituting (4.7) into (4.6), we get

$$\begin{aligned} \textrm{cap}_Q(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(x);G)\le C_1\left( \log \frac{1}{4\lambda }\left( e^{k_G(x,y)/(2c^2)} (1-1/e^{1/(2c^2)})\right) \right) ^{1-Q}. \end{aligned}$$

Again, by (4.2), we have

$$\begin{aligned} \textrm{cap}_Q(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(x);G)\le \tau k_G(x,y)^{1-Q}, \end{aligned}$$

where $\tau =(2c^2)^{Q-1}C_1.$ This proves (4.1), which indicates that G satisfies the $(\tau ,\lambda )$-k-cap condition.

$({\mathfrak {d}})\Rightarrow ({\mathfrak {a}})$. Assume that G is a Q-Loewner space and satisfies a $(\tau ,\lambda )$-k-cap condition, where $\tau >0$ and $0<\lambda <1/(16c)$. Since G is a Q-Loewner space, we know from Lemma 3.4 that G is Q-regular. Then it follows from [2, Proposition 6.48] that there are functions $\psi$: $(0, +\infty )\rightarrow (0, +\infty )$ and $\kappa$: $(0, +\infty )\rightarrow [1, +\infty )$ with the following property: If E and F are two disjoint nondegenerate continua in a ball ${{\mathbb {B}}}(x,r)$ in G, then

$$\begin{aligned} \textrm{cap}_Q(E,F; {{\mathbb {B}}}(x,\kappa (t)r))\ge \psi (t) \end{aligned}$$

(4.8)

whenever $\Delta (E,F)\le t$.

Let $t=16$ and $\varepsilon _0\in (0,\kappa (16)^{-1}]$. If E, F are disjoint nondegenerate continua in a ball ${{\mathbb {B}}}(x,\varepsilon d_G(x))$ in G with $0<\varepsilon \le \varepsilon _0$ and $\Delta (E,F)\le 16$, then we know from (4.8) that

$$\begin{aligned} \textrm{cap}_Q(E,F; {{\mathbb {B}}}(x,\kappa (16)\varepsilon d_G(x)))\ge \psi (16). \end{aligned}$$

Since G is c-quasiconvex, it follows from [2, Propositions 6.49] that G is locally Q-Loewner. This shows that, to prove this implication, we only need to check the uniformity of G. We are going to reach this goal by applying Theorem 2, that is, it suffices to show that there is a slow $\eta$ such that for any x, $y\in G$,

$$\begin{aligned} k_G(x,y)\le \eta (r_G(x,y)). \end{aligned}$$

(4.9)

To establish the existence of $\eta$, we consider two cases: $k_G(x,y)< 2$ and $k_G(x,y)\ge 2$. For the former, it follows from Lemma 2.1 (2) that

$$\begin{aligned} k_G(x,y)\le 3cr_G(x,y). \end{aligned}$$

(4.10)

For the latter, that is, $k_G(x,y)\ge 2$, since G is Q-regular and Q-Loewner, (2.5) in Theorem 5 guarantees that there is a decreasing self-homeomorphism $\psi$ of $(0,\infty )$ such that

$$\begin{aligned} \textrm{cap}_Q(E,F;G)\ge \psi (\Delta (E,F)) \end{aligned}$$

for all disjoint and nondegenerate continua E and F in G. Also, (2.6) in Theorem 5 ensures that there is a sufficiently large $\beta$ and a constant $C'>0$ satisfying that for all $t>\beta$,

$$\begin{aligned} \psi (t)\ge C'\left( \log t\right) ^{1-Q}. \end{aligned}$$

These show that for any pair of disjoint and nondegenerate continua E and F in G, if $\Delta (E,F)>\beta$, then

$$\begin{aligned} \textrm{cap}_Q(E,F;G)\ge C'\left( \log \Delta (E,F)\right) ^{1-Q}. \end{aligned}$$

(4.11)

Without loss of generality, we may assume that $\beta \ge 3$. The following assertion is useful for our arguments.

Claim 4.1

Suppose that $\lambda _1$ is a constant with $0<\lambda _1 \le \frac{1}{3(\beta +1)c}.$ Then for any continua $E_{1}\subset \lambda _1 {{\overline{{{\mathbb {B}}}}}}_G(x)$ and $F_{1}\subset \lambda _1 {{\overline{{{\mathbb {B}}}}}}_G(y)$, we have

$$\begin{aligned} \Delta (E_{1},F_{1})\ge \beta . \end{aligned}$$

Let $E_{1}\subset \lambda _1 {{\overline{{{\mathbb {B}}}}}}_G(x)$ and $F_{1}\subset \lambda _1 {{\overline{{{\mathbb {B}}}}}}_G(y)$ be continua. Obviously,

$$\begin{aligned} \Delta (E_{1},F_{1})\ge \Delta (\lambda _1\overline{{{\mathbb {B}}}}_G(x),\lambda _1{{\overline{{{\mathbb {B}}}}}}_G(y)). \end{aligned}$$

Since

$$\begin{aligned}{} & {} |x-y| \le {{\text {dist}}}(\lambda _1{{\overline{{{\mathbb {B}}}}}}_G(x),\lambda _1\overline{{{\mathbb {B}}}}_G(y))+\lambda _1 d_G(x)+\lambda _1 d_G(y)\\{} & {} \qquad \quad \le {{\text {dist}}}(\lambda _1{{\overline{{{\mathbb {B}}}}}}_G(x),\lambda _1\overline{{{\mathbb {B}}}}_G(y))+2\lambda _1\max \{ d_G(x), d_G(y)\}, \\{} & {} \max \{{{\text {diam}}}(\lambda _1{{\overline{{{\mathbb {B}}}}}}_G(x)), {{\text {diam}}}(\lambda _1\overline{{{\mathbb {B}}}}_G(y))\}\le 2\lambda _1\max \{d_G(x), d_G(y)\} \end{aligned}$$

and because Lemma 2.1() gives

$$\begin{aligned} |x-y|\ge \frac{2}{3c} \max \{d_G(x), d_G(y)\}, \end{aligned}$$

we get

$$\begin{aligned} \Delta (E_{1},F_{1})\ge \frac{1}{3c\lambda _1}-1\ge \beta , \end{aligned}$$

which is what we need.

Let

$$\begin{aligned} \lambda _2=\min \Big \{\lambda , \frac{1}{4(\beta +1)c}\Big \}. \end{aligned}$$

Then the assumptions $k_G(x,y)\ge 2$ and $\beta \ge 3$, together with Lemma 2.2, ensure that

$$\begin{aligned} \lambda _2 {{\overline{{{\mathbb {B}}}}}}_G(x)\cap \lambda _2 {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset , \end{aligned}$$

and so, Lemma 3.2 guarantees that there are two curves $\gamma _x\subset \lambda _2 {{\overline{{{\mathbb {B}}}}}}_G(x)$ and $\gamma _y\subset \lambda _2 {{\overline{{{\mathbb {B}}}}}}_G(y)$ such that both $\gamma _x$ and $\gamma _y$ are continua, and

$$\begin{aligned} \frac{1}{4\lambda _2}r_G(x,y)\le \Delta (\gamma _x,\gamma _y)\le \frac{3c}{2\lambda _2}r_G(x,y). \end{aligned}$$

(4.12)

Moreover, Claim 4.1 implies that

$$\begin{aligned} \Delta (\gamma _x,\gamma _y)\ge \beta \ge 3. \end{aligned}$$

(4.13)

Obviously, the fact $\lambda _2\le \lambda$ implies that $\gamma _x\subset \lambda {{\overline{{{\mathbb {B}}}}}}_G(x)$ and $\gamma _y\subset \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)$. By Theorem 3(3), we have

$$\begin{aligned} \textrm{cap}_Q(\gamma _x,\gamma _y;G)\le \textrm{cap}_Q(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y);G). \end{aligned}$$

Since $k_G(x,y)\ge 2$ and $\lambda {{\overline{{{\mathbb {B}}}}}}_G(x)\cap \lambda {{\overline{{{\mathbb {B}}}}}}_G(y)=\emptyset$ by Lemma 2.2, the assumption that G satisfies a $(\tau ,\lambda )$-k-cap condition indicates that

$$\begin{aligned} \textrm{cap}_Q(\lambda {{\overline{{{\mathbb {B}}}}}}_G(x),\lambda {{\overline{{{\mathbb {B}}}}}}_G(y);G)\le \frac{\tau }{k_G(x,y)^{Q-1}}, \end{aligned}$$

where $\tau >0$ and $0<\lambda \le 1/(16c)$. Thus we get

$$\begin{aligned} \textrm{cap}_Q(\gamma _x,\gamma _y;G)\le \frac{\tau }{k_G(x,y)^{Q-1}}. \end{aligned}$$

Because it follows from (4.11) and (4.13) that

$$\begin{aligned} \textrm{cap}_Q(\gamma _x,\gamma _y;G)\ge C' \big (\log \Delta (\gamma _x,\gamma _y)\big )^{1-Q}, \end{aligned}$$

we have

$$\begin{aligned} k_G(x,y)\le C_0\log \Delta (\gamma _x,\gamma _y), \end{aligned}$$

where $C_0=\big (\tau / C'\big )^{1/(Q-1)}$, and then, (4.12) leads to

$$\begin{aligned} k_G(x,y)\le C_0\log \left( \frac{3c}{2\lambda _2} r_G(x,y)\right) . \end{aligned}$$

(4.14)

To construct the needed control function $\eta$ in (4.9) based on (4.10) and (4.14), let

$$\begin{aligned} \eta _0(t)=\max \{\eta _1(t),\eta _2(t)\} \end{aligned}$$

for $t\in (0,\infty )$, where

$$\begin{aligned} \eta _1(t)=3ct\;\;\text{ and }\;\;\eta _2(t)=C_0\log (3ct/2\lambda _2). \end{aligned}$$

Since both $\eta _1$ and $\eta _2$ are strictly increasing homeomorphisms, by elementary computations, we see that $\eta _0$ is also a strictly increasing homeomorphism on $(0,+\infty )$ as well.

Since G is c-quasiconvex, [14, Theorem 2.7(1)] implies that for any x, $y\in G$,

$$\begin{aligned} k_G(x,y)\ge \log \big (1+r_G(x,y)\big ). \end{aligned}$$

This shows that if $r_G(x,y)\ge e^2-1$, then $k_G(x,y)\ge 2$. Let

$$\begin{aligned} t_0=e^2-1. \end{aligned}$$

Then we infer from the fact $\eta _2(t)\rightarrow +\infty$ as $t\rightarrow +\infty$ that there is $t_1\in (t_0,+\infty )$ such that $\eta _2(t_1)=2\eta _0(t_0)$. Let

$$\begin{aligned} \eta (t)={\left\{ \begin{array}{ll} \displaystyle \; \; \; \eta _0(t) &{} \text{ if }\quad t\in [0,t_0],\\ \; \; \; \max \{\eta _2(t),\eta _3(t)\} &{} \;\text{ if }\quad t\in (t_0,t_1],\\ \; \; \; \eta _2(t) &{} \;\text{ if }\quad t\in (t_1,+\infty ), \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \eta _3(t)=\frac{\eta _0(t_0)}{t_1-t_0}t +\frac{(t_1-2t_0)\eta _0(t_0)}{t_1-t_0}. \end{aligned}$$

Then $\eta$ is a strictly increasing homeomorphism. Moreover,

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{\eta (t)}{t}=\lim _{t\rightarrow +\infty } \frac{\eta _2(t)}{t}=0, \end{aligned}$$

which shows that the function $\eta$ is slow. Then we conclude from (4.10) and (4.14) that $\eta$ satisfies (4.9), and hence, it follows from Theorem 2 that the implication is true.$\square$

5 An Example

In this section, we shall provide two examples which show that the assumption of regularity in Theorem 1.1 can not be removed. Also, these examples demonstrate that for a metric measure space with Hausdorff dimension Q, it may not be Q-regular. Before the statement of these examples, let us introduce the following definitions.

Definition 5.1

A metric measure space $(X,d,\mu )$ is said to be doubling if there is a constant $C_d\ge 1$ such that

$$\begin{aligned} \mu ({{\mathbb {B}}}(x,2R))\le C_d\mu ({{\mathbb {B}}}(x,R)) \end{aligned}$$

for all balls ${{\mathbb {B}}}(x,R)$ in X of radius $0<R<{{\text {diam}}}X$. Also, we call $C_d$ a doubling constant of X.

Definition 5.2

Suppose that $(X,d,\mu )$ is a metric measure space and $1\le p< \infty$. We say that $(X,d,\mu )$ admits a weak (1, p)-Poincaré inequality if there are constants $\beta \ge 1$ and $C_{p}\ge 1$ such that for all balls ${{\mathbb {B}}}(x,R)$ in X with $R>0$,

whenever u is a bounded continuous function on $\beta {{\mathbb {B}}}(x,R)$ and $\rho$ is its very weak gradient. Also, $\beta$ is called a dilation constant of X. If $\beta =1$, then we say that $(X,d,\mu )$ admits a (1, p)-Poincaré inequality. We use the standard notation

for the mean value of a function u in a measurable set A of positive measure.

The following known results are useful for the arguments in this section.

Theorem 7

[11, Theorem 9.10]. Suppose that $(X,d,\mu )$ is a Q-regular, geodesic and proper metric measure space. Then $(X,d,\mu )$ admits a (1, Q)-Poincaré inequality if and only if it is a Q-Loewner space.

Theorem 8

[1, Theorem 4.4]. Assume that $\mu$ supports a weak (1, p)-Poincaré inequality on X, and let $G\subset X$ be an A-uniform domain. Then there exists $\alpha _0>0$ such that for each $\alpha \ge -\alpha _0$, the measure $d\nu (y)=d_{G}(y)^{\alpha }d\mu (y)$ is doubling on G and supports a weak (1, p)-Poincaré inequality on G with a dilation constant $\beta =3A$.

The following result is from [13, Proposition 2.2] and the proof of [13, Theorem 6.1].

Theorem 9

Fix $N\ge 3$. Consider points $x_0$, $x_1\in G\subset {{\mathbb {R}}}^n$ with $|x_0-x_1|\ge d_G(x_0)/N$, and let $\gamma$ be a quasihyperbolic geodesic in G joining $x_0$ and $x_1$. Then there exist a positive integer k with

$$\begin{aligned} \frac{N-1}{2}k_G(x_0,x_1)\le k\le 2Nk_G(x_0,x_1), \end{aligned}$$

(5.1)

and successive points $y_0=x_0, y_1, \ldots , y_k, y_{k+1}=x_1$ in $\gamma$ along the direction from $x_0$ to $x_1$ such that the closed balls $\{B_i={{\overline{{{\mathbb {B}}}}}}(y_i,r_i)\}$, where $r_i=d_G(y_i)/N$, satisfy that $\gamma \subset B_0\cup B_1\cup \cdots \cup B_{k+1}$, and for each $i\in \{0,1,\ldots , k\}$,

$$\begin{aligned} \frac{N-1}{N}r_i\le r_{i+1}\le \frac{N+1}{N}r_i\;\;\text{ and }\;\;B_{i+1}\subset \Big (2+\frac{1}{N}\Big )B_i\subset G. \end{aligned}$$

(5.2)

Now, we are ready to state and prove our first example.

Example 5.1

Suppose that $X=({{\mathbb {R}}}^2,|\cdot |,\mu )$, where $|\cdot |$ denotes the Euclidean metric, the measure $\mu$ is defined by $d\mu (x)=(1+|x|)dx$ and dx stands for the usual Lebesgue measure on ${{\mathbb {R}}}^2$. Let $G={{\mathbb {R}}}^2\setminus \{0\}$. Then the following statements are true.

• 1 X is doubling and lower 2-regular, but not 2-regular;

• 2 G is $\pi /\log 3$-uniform and $\pi /\log 3$-inner uniform;

• 3 G is a locally 2-Loewner and 2-Loewner space;

• 4 G satisfies a weak slice condition;

• 5 G does not satisfy any $(\tau ,\lambda )$-k-cap condition with $\tau >0$ and $0<\lambda \le 1/16$.

Proof

(1). The first statement of the example directly follows from [12, 5.3].

(2). By [10, Corollary 6.16], it follows that for each pair of points x, $y\in G$,

$$\begin{aligned} k_G(x,y)\le (\pi /\log 3)j_G(x,y), \end{aligned}$$

which implies that G is $\pi /\log 3$-uniform by [21, Theorem 6.16]. Obviously, G is $\pi /\log 3$-inner uniform as well.

(3). It is obvious that G is locally compact. Since $(G, |\cdot |,dx)$ is locally 2-Loewner by [2, Theorem 6.47], and since $d\mu \ge dx$, we see that $(G, |\cdot |, d\mu )$ is locally 2-Loewner as well. Similarly, we also know that $(G, |\cdot |, d\mu )$ is 2-Loewner.

(4). We already know that G is $\pi /\log 3$-uniform (see the statement (2) of the example), and also, it follows from [7, Page 202] that G is c-LEC for all $c>1$. Then Theorem 6 ensures that G satisfies a weak slice condition.

(5). Let $d\mu _1=dx$ and $d\mu _2=|x|dx$. Obviously, $d\mu =d\mu _1+d\mu _2$. Suppose on the contrary that G satisfies a $(\tau ,\lambda )$-k-cap condition, i.e., there are constants $0<\lambda <1/16$ and $\tau >0$ such that for each pair of points x, $y\in G$ with $k_G(x,y)\ge 2$,

$$\begin{aligned} \textrm{cap}_2(\lambda {{\overline{{{\mathbb {B}}}}}}(x),\lambda {{\overline{{{\mathbb {B}}}}}}(y);G)\le \frac{\tau }{k_G(x,y)}. \end{aligned}$$

(5.3)

$\square$

To reach a contradiction, we establish the following proposition.

Proposition 5.1

For each $t\in (1-e^{-2}, 1)$, there is a pair of points $x_t$, $y_t\in G$ with $k_G(x_t,y_t)\ge 2$ such that

$$\begin{aligned} \textrm{cap}_2(\lambda {{\overline{{{\mathbb {B}}}}}}(x_t),\lambda {{\overline{{{\mathbb {B}}}}}}(y_t);G)\ge C \end{aligned}$$

and

$$\begin{aligned} k_G(x_t,y_t)\rightarrow +\infty \ \ \text{ as } \ \ t\rightarrow 1^-, \end{aligned}$$

where $C>0$ is a constant and “$t\rightarrow 1^-$” means that $0<t<1$ and $t\rightarrow 1$.

For the proof, let N be an integer such that

$$\begin{aligned} N>\max \left\{ 65, \frac{1}{\lambda }\right\} . \end{aligned}$$

For each $t\in (1-e^{-2}, 1)$, let $x_t\in G$ with

$$\begin{aligned} |x_t|=\frac{2N^2t}{(N-1)(1-t)}, \end{aligned}$$

(5.4)

and $y_t=(1-t)x_t$. Then $y_t\in t{{\overline{{{\mathbb {B}}}}}}(x_t)$, and by [20, Lemma 2.2(1) and (2)],

$$\begin{aligned} \log \frac{1}{1-t}\le k_G(x_t,y_t)\le \frac{t}{1-t}. \end{aligned}$$

(5.5)

This implies that

$$\begin{aligned} k_G(x_t,y_t)\ge 2 \;\;\text{ and }\;\; k_G(x_t,y_t)\rightarrow +\infty \ \ \text{ as } \ \ t\rightarrow 1^-. \end{aligned}$$

(5.6)

Let

$$\begin{aligned} E_t=\frac{1}{N}{\overline{{{\mathbb {B}}}}}(x_t)\;\;\text{ and }\; \;F_t=\frac{1}{N}{\overline{{{\mathbb {B}}}}}(y_t). \end{aligned}$$

Since

$$\begin{aligned} \textrm{cap}_2(E_t,F_t;G)\le \textrm{cap}_2(\lambda \overline{{{\mathbb {B}}}}(x_t),\lambda {{\overline{{{\mathbb {B}}}}}}(y_t);G), \end{aligned}$$

it follows from (5.6) that to prove Proposition 5.1, we only need to show the following assertion.

Assertion 5.1

There is a constant $C>0$ such that

$$\begin{aligned} \textrm{cap}_2(E_t,F_t;G)\ge C. \end{aligned}$$

(5.7)

It follows from [10, Lemma 5.1] that there is a quasihyperbolic geodesic $\gamma$ in G joining $x_t$ and $y_t$. Since $t\ge 1-e^{-2}$ and $N>65$, we know that

$$\begin{aligned} |x_t-y_t|=t|x_t|\ge \frac{d_G(x_t)}{N}. \end{aligned}$$

(5.8)

To get the lower bound in (5.7), we need some preparation which consists of three steps.

Step 5.1

There is a finite sequence of closed balls $\{B_i\}$ in G such that their centers are in $\gamma$ and each $B_i$ satisfies (5.12) below.

It follows from (5.8) and Theorem 9 that there are an integer k and successive points $x_0=x_t,$ $x_1,$ $\ldots ,$ $x_k$, $x_{k+1}=y_t$ in $\gamma$ from $x_t$ to $y_t$ such that

$$\begin{aligned} \frac{N-1}{2}k_G(x_t,y_t)\le k\le 2Nk_G(x_t,y_t), \end{aligned}$$

(5.9)

and for each $0\le i\le k$,

$$\begin{aligned} \frac{N-1}{N}r_i\le r_{i+1}\le \frac{N+1}{N}r_i\;\;\text{ and }\;\; B_{i+1}\subset \frac{2N+1}{N}B_i, \end{aligned}$$

(5.10)

where $B_i={{\overline{{{\mathbb {B}}}}}}(x_i,r_i)$ and $r_i=d_G(x_i)/N$.

Obviously, $B_0=E_t$, $B_{k+1}=F_t$, and it follows from (5.6) and (5.9) that

$$\begin{aligned} k\ge 63. \end{aligned}$$

(5.11)

Since X is doubling, we get

$$\begin{aligned} \mu (B_i)\le \mu ((2+1/N)B_i)\le C_d\mu ((1+1/(2N))B_i)\le C^2_d\mu ((1/2+1/(4N))B_i), \end{aligned}$$

where $C_d$ denotes a doubling constant of X. Then it follows from the assumption $N\ge 65$ that

$$\begin{aligned} \mu (B_i)\le \mu ((2+1/N)B_i)\le C^2_d\mu (B_i). \end{aligned}$$

(5.12)

Step 5.2

For each $i\in \{0, 1,\ldots , k\}$, the quantities $\mu (B_i)$ and $\mu (B_{i+1})$ are equivalent. More precisely, we have

$$\begin{aligned} C'_0\mu (B_i)\le \mu (B_{i+1})\le C_0\mu (B_i), \end{aligned}$$

(5.13)

where

$$\begin{aligned} C_0=\max \left\{ \left( \frac{N+1}{N-1}\right) ^2 ,\frac{(N+1)^4}{N(N-1)^3}\right\} \end{aligned}$$

and

$$\begin{aligned} C'_0=\min \left\{ \left( \frac{N-1}{N+1}\right) ^2, \frac{(N-1)^2(N^2-2N-1)}{N(N+1)^3}\right\} . \end{aligned}$$

Let $i\in \{0, 1,\ldots , k\}$. Since $\mu _1(B_i)=\pi r^2_i$, it follows from (5.10) that

$$\begin{aligned} \left( \frac{N-1}{N+1}\right) ^2\mu _1(B_{i})\le \mu _1(B_{i+1})\le \left( \frac{N+1}{N-1}\right) ^2\mu _1(B_{i}). \end{aligned}$$

(5.14)

Next, we establish an equivalence between $\mu _2(B_{i})$ and $\mu _2(B_{i+1})$ as stated in (5.17) below. For this, let $\xi \in (2+1/N)B_i$. Then

$$\begin{aligned} d_G(x_i)-\frac{2N+1}{N}r_i\le |x_i|-|x_i-\xi |\le | \xi |\le |x_i|+|x_i-\xi |\le d_G(x_i)+\frac{2N+1}{N}r_i, \end{aligned}$$

which implies that

$$\begin{aligned} \frac{N^2-2N-1}{N^2}d_G(x_i)\le |\xi |\le \left( \frac{N+1}{N}\right) ^2d_G(x_i). \end{aligned}$$

Moreover, $B_{i+1}\subset (2+1/N)B_i$ by (5.10) and $\mu _2(B_{i+1})=\int _{B_{i+1}}|\xi |d\xi$. It follows that

$$\begin{aligned} \frac{N^2-2N-1}{N^2}d_G(x_i)\mu _1(B_{i+1})\le \mu _2(B_{i+1})\le \left( \frac{N+1}{N}\right) ^2d_G(x_i)\mu _1(B_{i+1}), \end{aligned}$$

and thus, (5.14) gives

$$\begin{aligned} \mu _2(B_{i+1})\le \frac{(N+1)^4}{N^2(N-1)^2}d_G(x_i)\mu _1(B_{i}) \end{aligned}$$

(5.15)

and

$$\begin{aligned} \mu _2(B_{i+1})\ge \frac{(N-1)^2(N^2-2N-1)}{N^2(N+1)^2}d_G(x_i)\mu _1(B_{i}). \end{aligned}$$

(5.16)

Moreover, for $\zeta \in B_i$, we have

$$\begin{aligned} d_G(x_i)-r_i=|x_i|-|\zeta -x_i|\le |\zeta |\le |\zeta -x_i|+|x_i|=r_i+d_G(x_i), \end{aligned}$$

which implies that

$$\begin{aligned} \frac{N-1}{N}d_G(x_i)\le |\zeta |\le \frac{N+1}{N}d_G(x_i). \end{aligned}$$

Thus we get

$$\begin{aligned} \frac{N}{(N+1)d_G(x_i)}\mu _2(B_i)\le \mu _1(B_i)\le \frac{N}{(N-1)d_G(x_i)}\mu _2(B_i). \end{aligned}$$

Now, we infer from (5.15) and (5.16) that

$$\begin{aligned} \frac{(N-1)^2(N^2-2N-1)}{N(N+1)^3}\mu _2(B_i)\le \mu _2(B_{i+1})\le \frac{(N+1)^4}{N(N-1)^3}\mu _2(B_i).\qquad \end{aligned}$$

(5.17)

Since $\mu =\mu _1+\mu _2$, we know from (5.14) and (5.17) that (5.13) is true.

Step 5.3

We establish an estimate on the quantity $\left| u_{B_{i+1}}-u_{B_i}\right|$ as follows: There is a constant $C_3>0$ such that for any admissible function u of the condenser $(E_t,F_t;G)$ and for each very weak gradient $\rho$ of u,

$$\begin{aligned} \left| u_{B_{i+1}}-u_{B_i}\right| \le C_3\left( \int _{\omega B_i}\rho ^2d\mu \right) ^{\frac{1}{2}} \end{aligned}$$

(5.18)

hold for all $i\in \{0,1,\ldots ,k\}$,where

$$\begin{aligned} \omega =3\pi (2N+1)/(N\log 3). \end{aligned}$$

Obviously,

$$\begin{aligned} \omega B_i\subset G, \end{aligned}$$

(5.19)

since $N\ge 65$.

For the proof, let $i\in \{0,1,\ldots ,k\}$, and let u be an admissible function for the condenser $(E_t,F_t;G)$. This means that

$$\begin{aligned} u|_{E_t}\ge 1\;\;\text{ and }\;\; u|_{F_t}\le 0. \end{aligned}$$

Since

$$\begin{aligned} \left| u_{B_{i+1}}-u_{B_i}\right| \le I_{i,1}+I_{i,2}, \end{aligned}$$

(5.20)

where $I_{i,1}=\left| u_{B_{i+1}}-u_{\frac{2N+1}{N}B_i}\right|$ and $I_{i,2}=\left| u_{B_{i}}-u_{\frac{2N+1}{N}B_i}\right|$, to estimate the quantity $\left| u_{B_{i+1}}-u_{B_i}\right|$, it suffices to estimate $I_{i,1}$ and $I_{i,2}$, respectively. We first work on the quantity $I_{i,1}$. Since

$$\begin{aligned} I_{i,1}\le \mu \left( B_{i+1}\right) ^{-1}\int _{B_{i+1}}\left| u-u_{\frac{2N+1}{N}B_i}\right| d\mu , \end{aligned}$$

we have

(5.21)

It is known that $({{\mathbb {R}}}^2, |\cdot |, \mu _1)$ is 2-regular, geodesic, proper and 2-Loewner. Then Theorem 7 implies that $({{\mathbb {R}}}^2, |\cdot |, \mu _1)$ admits a (1, 2)-Poincaré inequality. This ensures that there is a constant $C_{p,1}\ge 1$ such that for each very weak gradient $\rho$ of u,

(5.22)

As aforementioned, $({{\mathbb {R}}}^2, |\cdot |, \mu _1)$ admits a (1, 2)-Poincaré inequality. By the statement (2) of the example, G is $\pi /\log 3$-uniform. Then it follows from Theorem 8 that $(G, |\cdot |, \mu _2)$ admits a weak (1, 2)-Poincaré inequality with a dilation constant $\beta =3\pi /\log 3$. This means that there is a constant $C_{p,2}\ge 1$ such that for each very weak gradient $\rho$ of u,

(5.23)

To continue the estimate on $I_{i,1}$, we consider three cases according to the possible positions of $\omega B_i$ in X.

Claim 5.1

$\omega B_i\subset {{\mathbb {B}}}(0,1)$.

Under this assumption, we get

$$\begin{aligned} d\mu _1\le d\mu \le 2d\mu _1 \end{aligned}$$

on $\omega B_i$. Then

and so, we get

(5.24)

Claim 5.2

$\omega B_i\subset X\setminus {{\mathbb {B}}}(0,1)$.

Under this assumption, we get

$$\begin{aligned} d\mu _2\le d\mu \le 2d\mu _2 \end{aligned}$$

on $\omega B_i$. Then

and thus, we have

(5.25)

since $\mu _2(\omega B_i)\ge 1/2\mu (\omega B_i)\ge 1/2\mu _1(\omega B_i)$.

Claim 5.3

$\omega B_i\cap {{\mathbb {B}}}(0,1)\ne \emptyset$ and $\omega B_i\cap (X\setminus {{\mathbb {B}}}(0,1))\ne \emptyset$.

Proof

Let

Then

(5.26)

since $d\mu _1\le d\mu$ and $d\mu _2\le d\mu$.

It follows from the fact $\mu _1(((2N+1)/N)B_i)=\pi ((2N+1)r_i/N)^2$ and (5.22) that

$$\begin{aligned} {\mathfrak {T}}_1\le \frac{N C_{p,1}}{(2N+1)\pi ^{\frac{1}{2}}}\left( \int _{\frac{2N+1}{N}B_i} \rho ^2d\mu _1\right) ^{\frac{1}{2}}. \end{aligned}$$

Thus we get

$$\begin{aligned} {\mathfrak {T}}_1\le & {} \frac{N C_{p,1}}{(2N+1)\pi ^{\frac{1}{2}}}\left( \int _{\omega B_i}\rho ^2d\mu _1\right) ^{\frac{1}{2}}\,\;\;\; {(\text{ since }\quad ((2N+1)/N)B_i\subset \omega B_i)}\nonumber \\\le & {} \frac{N C_{p,1}}{(2N+1)\pi ^{\frac{1}{2}}}\left( \int _{\omega B_i}\rho ^2d\mu \right) ^{\frac{1}{2}}.\;\;\;\; {(\text{ since }\quad d\mu _1\le d\mu )} \end{aligned}$$

(5.27)

To estimate the quantity ${\mathfrak {T}}_2$, we need some preparation. Let $z_1\in {{\mathbb {B}}}(0,1)\cap \omega B_i$. Then we get

$$\begin{aligned} |x_i|\le |x_i-z_1|+|z_1|<\frac{\omega }{N}|x_i|+1, \end{aligned}$$

which implies that

$$\begin{aligned} |x_i|<\frac{N}{N-\omega }. \end{aligned}$$

Let $z_2\in \left( X\setminus {{\mathbb {B}}}(0,1)\right) \cap \omega B_i$. Then we have

$$\begin{aligned} |x_i|\ge |z_2|-|x_i-z_2|>1-\frac{\omega }{N}|x_i|, \end{aligned}$$

which ensures that

$$\begin{aligned} |x_i|>\frac{N}{N+\omega }. \end{aligned}$$

These show that

$$\begin{aligned} \frac{N}{N+\omega }<|x_i|<\frac{N}{N-\omega }, \end{aligned}$$

and thus,

$$\begin{aligned} \frac{1}{N+\omega }<r_i<\frac{1}{N-\omega }, \end{aligned}$$

since $r_i=|x_i|/N$. Now, we conclude that for any $x\in \omega B_i$,

$$\begin{aligned} \frac{N^2-2N\omega -\omega ^2}{N^2-\omega ^2}{} & {} <|x_i|-|x-x_i|\le |x|\le |x-x_i|+|x_i|\nonumber \\{} & {} < \frac{N+\omega }{N-\omega }. \end{aligned}$$

(5.28)

We are ready to estimate ${\mathfrak {T}}_2$. Since (5.23) implies

$$\begin{aligned} {\mathfrak {T}}_2\le \frac{C_{p,2}r_i}{\mu _2(\omega B_i)^{\frac{1}{2}}}\left( \int _{\omega B_i}\rho ^2d\mu _2\right) ^{\frac{1}{2}}, \end{aligned}$$

and since (5.28) gives

$$\begin{aligned} \mu _2(\omega B_i)=\int _{\omega B_i}|z|dz\ge Kr^2_i, \end{aligned}$$

where $K=\frac{\pi \omega ^2(N^2-2N\omega -\omega ^2)}{N^2-\omega ^2}$, we obtain that

$$\begin{aligned} {\mathfrak {T}}_2 \le \frac{C_{p,2}}{K^{\frac{1}{2}}}\left( \int _{\omega B_i}\rho ^2d\mu _2\right) ^{\frac{1}{2}} \le \frac{C_{p,2}}{K^{\frac{1}{2}}}\left( \int _{\omega B_i}\rho ^2d\mu \right) ^{\frac{1}{2}}, \end{aligned}$$

(5.29)

where, in the second inequality, the fact $d\mu _2\le d\mu$ is applied.

Let

$$\begin{aligned} K'=\frac{N C_{p,1}}{(2N+1)\pi ^{\frac{1}{2}}} +\frac{C_{p,2}}{K^{\frac{1}{2}}}. \end{aligned}$$

Then we conclude from (5.26), (5.27) and (5.29) that

(5.30)

Let

$$\begin{aligned} M=\max \left\{ \frac{6\sqrt{\pi } C_{p,1}}{\omega \log 3}, \frac{2\sqrt{2}C_{p,2}}{\pi ^{\frac{1}{2}}\omega }, K'\right\} . \end{aligned}$$

Then the combination of (5.24), (5.25) and (5.30) ensures that

(5.31)

and so, we get from (5.21) that

$$\begin{aligned} I_{i,1}\le H\left( \int _{\omega B_i}\rho ^2d\mu \right) ^{\frac{1}{2}}, \end{aligned}$$

(5.32)

where $H=MC^2_d/C'_0$. This is an upper bound on $I_{i,1}$ which we need.

Next, we work on the quantity $I_{i,2}$. Since

$$\begin{aligned} I_{i,2}=\left| \mu \left( B_{i}\right) ^{-1}\int _{B_{i}} \left( u-u_{\frac{2N+1}{N}B_i}\right) d\mu \right| , \end{aligned}$$

we have that

(5.33)

where $H'=MC^2_d$. This is an upper bound on $I_{i,2}$ which we need.

By letting $C_3=H+H'$, we know from (5.20), (5.32) and (5.33) that (5.18) is true.

In the following, based on the relation (5.18), we are going to check the estimate (5.7) in Assertion 5.1.

First, we need a lower bound for |x| for any $x\in E_t=(1/N){{\overline{{{\mathbb {B}}}}}}(x_t)$. Let $x\in E_t$. Then it follows from (5.4) that

$$\begin{aligned} |x|\ge |x_t|-|x_t-x|\ge \frac{2N^2t}{(N-1)(1-t)} -\frac{2Nt}{(N-1)(1-t)}=\frac{2Nt}{1-t}. \end{aligned}$$

Since $u|_{E_t}\ge 1$ and $u|_{F_t}\le 0$, we have that

and so, we obtain from (5.5) and (5.9) that

$$\begin{aligned} k\le 2Nk_G(x_t,y_t)\le \frac{2Nt}{1-t}\le u_{E_t}-u_{F_t}. \end{aligned}$$

(5.34)

Moreover, by (5.18), we get

$$\begin{aligned} u_{E_t}-u_{F_t}\le \sum _{i=0}^{k}|u_{B_{i+1}}-u_{B_i}|\le C_3(k+1)^{\frac{1}{2}}\left( \sum _{i=0}^{k}\int _{\omega B_i}\rho ^2d\mu \right) ^{\frac{1}{2}}. \end{aligned}$$

(5.35)

Then we deduce that

$$\begin{aligned} k\le & {} {} C_3(k+1)^{\frac{1}{2}}\left( \sum _{i=0}^{k}\int _{\omega B_i}\rho ^2d\mu \right) ^{\frac{1}{2}}\;\;\;(\text {by }\, (5.34)\, \text { and } \,(5.35))\nonumber \\\le & {} {} C_3(k+1)\left( \int _G \rho ^2d\mu \right) ^{\frac{1}{2}}. \;\;\;\;\;\;\;\;\;\;\;\;\;(\text {by }\,(5.19)) \end{aligned}$$

(5.36)

It follows from (5.11) that $1+k\le 2k$, and thus, we infer from (5.36) that

$$\begin{aligned} \int _G \rho ^2d\mu \ge \Big (\frac{1}{2C_3}\Big )^2=C, \end{aligned}$$

from which the estimate (5.7) in Assertion 5.1 follows. Hence Proposition 5.1 is proved as well.

Now, we are ready to get a contradiction. Since (5.6) shows that for each $t\in (1-e^{-2}, 1)$, $k_G(x_t,y_t)\ge 2$, we know from (5.3) and Proposition 5.1 that

$$\begin{aligned} C\le \frac{\tau }{k_G(x_t,y_t)} \end{aligned}$$

hold for all $t\in (1-e^{-2}, 1)$. This is impossible since (5.6) ensures that $k_G(x_t,y_t)\rightarrow +\infty$ as $t\rightarrow 1^-$. This contradiction shows that G does not satisfy any $(\tau ,\lambda )$-k-cap condition. $\square$

Similar arguments as in the proof of Example 5.1 guarantee that the following example is true as well.

Example 5.2

Suppose that $X=({{\mathbb {R}}}^2\setminus \{0\},|\cdot |,\mu )$, where $|\cdot |$ denotes the Euclidean metric, the measure $\mu$ is defined by $d\mu (x)=(1+|x|)dx$, and dx stands for the usual Lebesgue measure on ${{\mathbb {R}}}^2$. Then the following statements are true.

• (1) X is doubling and lower 2-regular, but not 2-regular;

• (2) X is $\pi /\log 3$-uniform and $\pi /\log 3$-inner uniform;

• (3) X is a locally 2-Loewner and 2-Loewner space;

• (4) X satisfies a weak slice condition;

• (5) X does not satisfy any $(\tau ,\lambda )$-k-cap condition with $\tau >0$ and $0<\lambda \le 1/16$.

We end this paper with the following problem: Under the assumptions of Theorem 1.1, are the following statements quantitatively equivalent: (1) G is uniform and locally Q-Loewner; (2) G is Q-Loewner and Gromov hyperbolic?